Ayan created these clues to discribe her numbers.She used each of the 10 digits 0to 9 once to make the two numbers.
It is between 52000 and 2000.
there is a 2 in the tenth place.
the sum of the digits is 23
To find the two numbers Ayan created, we can follow these steps:
Step 1: Determine the possibilities for the tens digit
The clue states that there is a 2 in the tenth place, meaning the tens digit can only be 2.
Step 2: Find the possibilities for the units digit
To find the units digit, we need to subtract the sum of the digits from 23. Since the sum of the digits is 23, the difference between 23 and the sum of the tens and units digits would give us the value of the units digit.
Let's assume the units digit is x. Then we have the equation: 2 + x = 23 - sum of the other digits
Simplifying, we get: x = 21 - sum of the other digits
Step 3: Find the remaining digits
Since we know that the number is between 52000 and 2000, we can determine the possible range for the remaining digits. The highest value the units digit (x) can be is 9, giving us the highest possible number of 52999. The lowest value for x would be 0, resulting in the lowest possible number of 52000.
Step 4: Determine the final number
We have the tens digit as 2 and the units digit as x. We need to distribute the remaining digits between the thousands, hundreds, and tens places while making sure to use each of the remaining digits (0, 1, 3, 4, 5, 6, 7, 8, 9) exactly once.
By examining the remaining digits, we can narrow down the possible values for the tens, hundreds, and thousands digits. The highest remaining digit is 9, so it must be placed in the thousands place to create a number higher than 2000. The next highest remaining digit is 8, which must then be placed in the hundreds place. This leaves us with the digits 0, 1, 3, 4, 5, 6, and 7 for the tens place.
We can now try different combinations to find the two numbers that meet the given conditions:
- Starting with the thousands digit: 9 (highest remaining digit)
- Next, the hundreds digit: 8 (second-highest remaining digit)
- Then, the tens digit: Try each of the remaining digits one by one: 0, 1, 3, 4, 5, 6, 7
By trying each of the remaining digits for the tens place, we can find the two numbers that meet the given conditions:
If 0 is the tens digit:
- Thousands digit: 9
- Hundreds digit: 8
- Units digit: 2
The first number would be 9802.
If 1 is the tens digit:
- Thousands digit: 9
- Hundreds digit: 8
- Units digit: 5
The second number would be 9852.
Thus, the two numbers Ayan created are 9802 and 9852.